Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(x,y):\mathrm{Aff}/U \to \mathrm{Ens}$ is represented by an algebraic $U$-space, for every $U \in \mathrm{Aff}/S$ and all $x,y \in \mathcal{X}_U$.

I've seen this shown by claiming that the stack associated to $\mathcal{Isom}(x,y)$ is canonically isomorphic to $U \times_{(x,y),\mathcal{X} \times_S \mathcal{X}, \Delta} \mathcal{X}$. My question is how does one make this identification?